1. Field of the Invention
The invention generally relates to a system and method for calculating image profiles on a surface and, more particularly, to a method and system to calculate estimated image profiles resulting from imaging in the presence of lens aberrations.
2. Background Description
The calculation of image profiles in the presence of lens aberrations has become a routine requirement. Image profiles come in many forms, with the aerial image profile being the most fundamental. The aerial image profile is defined as a description of the variation of optical energy (e.g. represented as “amplitude” or “intensity”) with physical location within or nearby some plane being considered, herein defined as the “image plane”. This optical image profile is known to impact all subsequent forms of image profile, such as the image profile within the developed photoresist, and the further subsequent image profile etched into a process layer on the wafer. Hence, the ability to simulate the impact of lens aberrations on the aerial image profile is a key requirement to being able to simulate any other image profile associated with later steps in the semiconductor manufacturing process.
These calculations of simulated aerial image profiles are important in the field of photolithography, especially in view of the decreased size of printed dimensions on a wafer surface. By way of example, technologies have reached beyond the 100 nm scale achieving, in some instances, over (100,000,000) transistors on a single wafer chip.
But to achieve such densities, it is important, for example, to adjust the projection lenses within the projection lithography tools for lens aberrations at several steps within the lens fabrication process, as well as during final installation at the end-user's wafer fabrication facility. This lens adjustment process may make use of a simulation calculation. This will ensure that “adjustment state” of a lens, as represented by the measurements of its lens aberration content, is optimum for the imaging of a particular specified target optimization pattern. Thus, adjustments to the lens can be prescribed and made throughout the fabrication process, and during the installation at the customer facility as well, while ensuring that the delivered image profile is within design parameters.
Many simulation packages are available commercially, but these simulation packages are exceptionally computationally intensive and thus are not very practical in the lens adjustment application described above. This is primarily due to the fact that the above typical lens adjustment application of image simulation would require new simulation calculations to be executed for each and every considered change of any one or more lens element positions. Since the mathematical optimization routines typically applied utilize many thousands of “trial adjustments” while they hunt for the best possible combination of all adjustments, this translates to many thousands of required image simulation calculations, if the image profile is to be used as a metric to evaluate the “goodness” of each considered lens adjustment combination.
A typical current image simulation technique, as described in more detail below, uses a very complex calculation for simulating an image profile on the wafer. However, with this simulation technique, a new calculation of a propagated aerial image is required each and every time a different set of aberration component values (most typically represented as Zernike Polynomial coefficient values, in current practice) is to be considered by the simulation environment. That is, for each set of new aberration components for which a corresponding image profile is desired:                (i) a mathematical series (or the coefficients of an agreed-upon series) must be provided as input, describing the aberrations in the lens when imaging at a single exposure field position to be considered;        (ii) an aberration calculation is performed for each new set of aberration components (i.e., the value of the series representing the aberration is calculated at each position within the pupil plane of simulated lens being considered); and        (iii) a final image simulation calculation is performed using the new pupil-plane aberration representation.        
But, this sort of “full” image simulation calculation is very time consuming and mathematically intensive. And since, for example, in the lens adjustment application mentioned above, many thousands of different possible lens adjustment trials will be considered, and for each trial lens adjustment many different positions within the exposure field (each with an independent aberration condition/series) must be considered, it is clear that executing full image simulations at each field point for every trial lens adjustment would not be efficient enough to be practical. If it is desired to use the simulated image profile as a metric to judge and rank the “goodness” of one particular lens adjustment state to another, a faster calculation of image profiles is necessary.
Partially Coherent Imaging Theory as typically applied in the field of photolithography image simulation is summarized below, as an illustration of the specific steps required in the current practice version of atypical calculation of an aberrated image profile. Specifically, to date the theoretical analysis of projection imaging (e.g. Koehler matched illumination, diffraction-limited projection lens, etc., as in photolithography) has primarily treated the illumination source as “quasi-monochromatic,” with each point in the illumination source yielding the image of any considered object-point “coherently”. The field of partial coherence has developed primarily as a study of the variation in imaging with a variation in the distribution of such illumination source-points.
By way of example, each illuminator source-point yields a spatial frequency spectrum on the projection lens entrance pupil sphere which is slightly shifted (in space) with respect to the spectrum arriving from a neighboring source-point. The result is that a fixed-position pupil-aperture passes a different “chunk” of the total spatial frequency spectrum emitted per object (for each illuminator source-point considered). In other words, this is one of the reasons for aberrations, which will impact on the image profile. And, when applying this method (or those related), each consideration of a new set of aberration coefficients requires a new mathematical treatment of applying the phase shift induced by the aberrations prior to the calculation of the final image profile. Under prior art which utilized such methods, this recalculation of the propagation of light from the pupil plane of the lens to the image plane is indeed executed each time a new or different set of aberration coefficients are to be considered.
As described above, to account for the many problems associated with the lens and light source, current imaging models sum the “intensity-image” contributions from each illumination source-point to build the final image irradiance (“intensity”) profile. A typical functional representation of a partially coherent image intensity begins with:
                                          1            )                    ⁢                                          ⁢                      I            ⁡                          (                                                r                  →                                ,                z                            )                                      =                                            ∫              sourceS                        ⁢                                                  ⁢                                          ⅆ                                  ρ                  0                                            ⁢                              J                ⁡                                  (                                                            ρ                      →                                        0                                    )                                            ⁢                                                                                      E                    ⁡                                          (                                                                        r                          →                                                ,                                                  z                          ;                                                                                    ρ                              →                                                        0                                                                                              )                                                                                        2                                              =                                    ∫              sourceS                        ⁢                                                  ⁢                                          ⅆ                                  ρ                  0                                            ⁢                              J                ⁡                                  (                                                            ρ                      →                                        0                                    )                                            ⁢                                                I                  c                                ⁡                                  (                                                            r                      →                                        ,                                          z                      ;                                                                        ρ                          →                                                0                                                                              )                                                                                                    where,
E is the electric field whose squared modulus becomes the intensity, both of which are shown to be a function of the source point at position {right arrow over (ρ)}0;
J({right arrow over (ρ)}0) is the effective illumination source distribution in the lens pupil, so in effect defines the attenuation of amplitude and/or shift of initial phase per source point being considered;
Ic is the coherent image from the single source point;
{right arrow over (r)} defines the lateral local image position vector (i.e. x/y position) in the image plane; and
z is the axial image position.
Further expanding the electric field portion:
                                          2            )                    ⁢                                          ⁢                      I            ⁡                          (                                                r                  →                                ,                z                            )                                      =                              ∫            sourceS                    ⁢                                          ⁢                                    ⅆ                                                ρ                  →                                0                                      ⁢                          J              ⁡                              (                                                      ρ                    →                                    0                                )                                      ⁢                                                  ⁢                                                                                                ∫                                          ≥                      ExitPupil                                                        ⁢                                                                          ⁢                                                            ⅆ                                              ρ                        →                                                              ⁢                                                                  O                        ~                                            ⁡                                              (                                                                              ρ                            →                                                    -                                                                                    ρ                              →                                                        0                                                                          )                                                              ⁢                                          P                      ⁡                                              (                                                  ρ                          →                                                )                                                              ⁢                                          F                      ⁡                                              (                                                                              ρ                            →                                                    ,                          z                                                )                                                              ⁢                                          ⅇ                                                                        -                                                      ik                            0                                                                          ⁢                                                                              r                            →                                                    ·                                                      ρ                            →                                                                                                                ⁢                                          ⅇ                                                                        -                                                      ik                            0                                                                          ⁢                                                  z                          0                                                ⁢                        γ                                                              ⁢                                          ⅇ                                                                        -                                                      ik                            0                                                                          ⁢                                                  W                          ⁡                                                      (                                                          ρ                              →                                                        )                                                                                                                                                                          2                                                                                                                              3              )                        ⁢                                                  ⁢                          I              ⁡                              (                                                      r                    →                                    ,                  z                                )                                              =                                    ∫              sourceS                        ⁢                                                  ⁢                                          ⅆ                                  ρ                  0                                            ⁢                              J                ⁡                                  (                                                            ρ                      →                                        0                                    )                                            ×                              ∫                                                      ∫                                          ≥                      ExitPupil                                                        ⁢                                                                          ⁢                                                            ⅆ                                              ρ                        →                                                              ⁢                                          ⅆ                                                                                          ⁢                                                                        ρ                          →                                                ′                                                              ⁢                                                                  O                        ~                                            ⁡                                              (                                                                              ρ                            →                                                    -                                                                                    ρ                              →                                                        0                                                                          )                                                              ⁢                                          O                      ~                                        *                                          (                                                                                                    ρ                            →                                                    ′                                                -                                                                              ρ                            →                                                    0                                                                    )                                        ⁢                                          P                      ⁡                                              (                                                  ρ                          →                                                )                                                              ⁢                                                                  P                        *                                            ⁡                                              (                                                                              ρ                            →                                                    ′                                                )                                                              ⁢                                          F                      ⁡                                              (                                                                              ρ                            →                                                    ,                          z                                                )                                                              ⁢                                                                                            F                          *                                                ⁡                                                  (                                                                                                                    ρ                                →                                                            ′                                                        ,                            z                                                    )                                                                    ·                                              ⅇ                                                                              -                                                          ik                              0                                                                                ⁢                                                                                    r                              →                                                        ·                                                          (                                                                                                ρ                                  →                                                                -                                                                                                      ρ                                    →                                                                    ′                                                                                            )                                                                                                                                            ⁢                                          ⅇ                                                                        -                                                      ik                            0                                                                          ⁢                                                                              z                            0                                                    ⁡                                                      (                                                          γ                              -                                                              γ                                ′                                                                                      )                                                                                                                ⁢                                          ⅇ                                              -                                                                              ik                            0                                                    ⁡                                                      [                                                                                          W                                ⁡                                                                  (                                  ρ                                  )                                                                                            -                                                              W                                ⁡                                                                  (                                                                      ρ                                    →                                                                    )                                                                                                                      ]                                                                                                                                                                                  ⁢                                                                  In Equations (2) and (3), the double integral accounts for the squaring process, with integration being taken over the lens “exit pupil transmission function” (the explicit inclusion of the lens transmission function relaxes the requirements on the limits of integration, with the result that the limits need only be greater than or equal to the region of space containing the lens aperture). Additionally,
Õ({right arrow over (ρ)}) is the Fourier Transform of the object transmission function;
P({right arrow over (ρ)}) is the lens transmission pupil function which is generally a one/zero real-valued function (but it is not forced to be so); and
F({right arrow over (ρ)},z) is the modification of the aerial image when considered in a thin film (required when modeling an image in photoresist).
Also, the exponential terms describe the propagation of the fields according to pupil position, with the first exponential containing the x-y functionality, the second exponential handling the z-propagation, and the last exponential mapping the phase aberration from a reference sphere in the lens exit pupil.
When viewed in conjunction with the first integral, the shifted version of the object's spatial frequency spectrum Õ({right arrow over (ρ)}) combines to yield a convolution of it with the effective illumination source distribution. It is the nature of this effective source distribution function J({right arrow over (ρ)}0) which drives whether any mutual interaction among different effective source points will exist or not. As discussed, in all current photolithography imaging models, the squared modulus of the electric field is taken for each source point considered, and the sum of these “illuminator source-point-based intensity profiles” is taken as the total image. The justification for this relates to the nature of the source. However, the very fundamentals of this sort of calculation technique demands recalculation of the light's path from the pupil of the projection lens to the image plane each and every time a change is made in the aberration content of the lens.
In the current practice of image simulation, commercial software packages exist that provide the capability of “image profile estimation” (e.g., Prolith™ by KLA-Tencor). The core function that these software routines provide is the calculation of a propagated aerial image, with the optional follow-up calculation of a “developed image profile in resist”. Said otherwise, these software routines may calculate (i) the image profile transmitted by a lens which contains an input description of its aberration content (e.g., a set of Zernike coefficients), and (ii) the resultant image profile in the resist (either before or after development of the resist).
All such software programs, however, require a new calculation of a propagated aerial image each and every time a different set of aberration component values is to be considered. As discussed, this aberration calculation is necessarily complicated and still very time consuming. And when considered in light of applications which may require many tens or even hundreds of thousands of full image simulation calculations, such as the lens adjustment application described previously, it is seen that the application of such currently available software to this task would be prohibitive to achieving a result within allotted time constraints.
The invention is directed to overcoming one or more of the problems as set forth above.